Optimal. Leaf size=72 \[ -\frac {(A b-a C) \log \left (a+b x^2\right )}{2 a b}+\frac {A \log (x)}{a}+\frac {(b B-a D) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}+\frac {D x}{b} \]
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Rubi [A] time = 0.10, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1802, 635, 205, 260} \[ -\frac {(A b-a C) \log \left (a+b x^2\right )}{2 a b}+\frac {A \log (x)}{a}+\frac {(b B-a D) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}+\frac {D x}{b} \]
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 635
Rule 1802
Rubi steps
\begin {align*} \int \frac {A+B x+C x^2+D x^3}{x \left (a+b x^2\right )} \, dx &=\int \left (\frac {D}{b}+\frac {A}{a x}+\frac {a (b B-a D)-b (A b-a C) x}{a b \left (a+b x^2\right )}\right ) \, dx\\ &=\frac {D x}{b}+\frac {A \log (x)}{a}+\frac {\int \frac {a (b B-a D)-b (A b-a C) x}{a+b x^2} \, dx}{a b}\\ &=\frac {D x}{b}+\frac {A \log (x)}{a}-\frac {(A b-a C) \int \frac {x}{a+b x^2} \, dx}{a}+\frac {(b B-a D) \int \frac {1}{a+b x^2} \, dx}{b}\\ &=\frac {D x}{b}+\frac {(b B-a D) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}+\frac {A \log (x)}{a}-\frac {(A b-a C) \log \left (a+b x^2\right )}{2 a b}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 73, normalized size = 1.01 \[ \frac {(a C-A b) \log \left (a+b x^2\right )}{2 a b}+\frac {A \log (x)}{a}-\frac {(a D-b B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}+\frac {D x}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 158, normalized size = 2.19 \[ \left [\frac {2 \, D a b x + 2 \, A b^{2} \log \relax (x) - {\left (D a - B b\right )} \sqrt {-a b} \log \left (\frac {b x^{2} + 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + {\left (C a b - A b^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, a b^{2}}, \frac {2 \, D a b x + 2 \, A b^{2} \log \relax (x) - 2 \, {\left (D a - B b\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + {\left (C a b - A b^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, a b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 66, normalized size = 0.92 \[ \frac {D x}{b} + \frac {A \log \left ({\left | x \right |}\right )}{a} - \frac {{\left (D a - B b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b} + \frac {{\left (C a - A b\right )} \log \left (b x^{2} + a\right )}{2 \, a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 80, normalized size = 1.11 \[ \frac {B \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}-\frac {D a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b}+\frac {A \ln \relax (x )}{a}-\frac {A \ln \left (b \,x^{2}+a \right )}{2 a}+\frac {C \ln \left (b \,x^{2}+a \right )}{2 b}+\frac {D x}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.93, size = 65, normalized size = 0.90 \[ \frac {D x}{b} + \frac {A \log \relax (x)}{a} - \frac {{\left (D a - B b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b} + \frac {{\left (C a - A b\right )} \log \left (b x^{2} + a\right )}{2 \, a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {A+B\,x+C\,x^2+x^3\,D}{x\,\left (b\,x^2+a\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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